Question: The sum of the first $3$ terms of a geometric series is $171$ and the common ratio is $\dfrac23$. What is the first term of the series?
This formula gives the sum ${S_n}$ of the first $ n$ terms in the geometric series where the first term is $ a$ and the common ratio is $C r$ : ${S_n}=\dfrac{ a(1-C r^{ n})}{1-C r}$ We are given the values for ${S_n}$, $ n$, and $C r$.Let's plug them in the formula and solve for $ a$. We are given that ${S_n=171}$, ${n=3}$, and $C{r=\dfrac23}$ : ${171}=\dfrac{ a\left(1-\left(C{\dfrac23}\right)^{{3}}\right)}{1-\left(C{\dfrac23}\right)}$ Solving the equation, we get that $a=81$. In conclusion, the first term of the series is $81$.